Remark 43.15.6 (Trivial generalization). Let (A, \mathfrak m, \kappa ) be a Noetherian local ring. Let M be a finite A-module. Let I \subset A be an ideal. The following are equivalent
I' = I + \text{Ann}(M) is an ideal of definition (Algebra, Definition 10.59.1),
the image \overline{I} of I in \overline{A} = A/\text{Ann}(M) is an ideal of definition,
\text{Supp}(M/IM) \subset \{ \mathfrak m\} ,
\dim (\text{Supp}(M/IM)) \leq 0, and
\text{length}_ A(M/IM) < \infty .
This follows from Algebra, Lemma 10.62.3 (details omitted). If this is the case we have M/I^ nM = M/(I')^ nM for all n and M/I^ nM = M/\overline{I}^ nM for all n if M is viewed as an \overline{A}-module. Thus we can define
and we get
for all n by the equalities above. All the results of Algebra, Section 10.59 and all the results in this section, have analogues in this setting. In particular we can define multiplicities e_ I(M, d) for d \geq \dim (\text{Supp}(M)) and we have
as in the case where I is an ideal of definition.
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